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This article is cited in 3 scientific papers (total in 3 papers)
Transient phenomena in a random walk
A. K. Aleshkyavichene, S. V. Nagaev Institute of Mathematics and Informatics
Abstract:
The paper studies the limit distributions of the maximum of sums
$\max_{1\le k\le n}\sum_{l=1}^k\xi_{n,l}$ for the triangular array $\xi_{n,k}$,
$k=1,\ldots,n$, $n=1,2,\ldots $, of independent identically distributed
random variables in a singular series in cases where
$a_n=E\xi_{n,k}\to
0$ and/or 1) $a_n\sqrt n\to\infty$, or 2) $a_n\sqrt n\to-\infty$,
or 3) $a_n\sqrt n\to 0$ as $n\to\infty$.
The direct proof that the analytic expressions for limit laws
coincide was previously obtained
by different authors and is given. Moreover,
for these transient cases the convergence of the sequence of distributions
of maximums to the limit laws is proved with the help of the characteristic
functions method.
Keywords:
triangular array, maximum of sequential sums, limit distributions, method of characteristic functions.
DOI:
https://doi.org/10.4213/tvp298
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English version:
Theory of Probability and its Applications, 2004, 48:1, 1–18
Bibliographic databases:
Received: 17.11.1998
Citation:
A. K. Aleshkyavichene, S. V. Nagaev, “Transient phenomena in a random walk”, Teor. Veroyatnost. i Primenen., 48:1 (2003), 3–21; Theory Probab. Appl., 48:1 (2004), 1–18
Citation in format AMSBIB
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Linking options:
http://mi.mathnet.ru/eng/tvp298https://doi.org/10.4213/tvp298 http://mi.mathnet.ru/eng/tvp/v48/i1/p3
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